# What limits the maximum sustainable surface charge density of a sphere in space?

Suppose I charge a sphere and leave it in vacuum for 10 years. After that time, I want its surface charge density to be in the order of 10^5C/m^2. Would that be possible? Would it depend on the material used and how? Would adding or removing electrons make a difference (positively vs negatively charged)?

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Why would there be a limit in vacuum? – ABC Apr 27 '13 at 5:03
Thank you for your answer. I kind of see your point, if it is a hypothetical, perfect sphere. I don't suppose that even in that case we could add an infinite number of electrons, but maybe I am wrong. My question though, is about a sphere that we could actually build. Wouldn't it necessarily have imperfections that would allow electrons to escape? – Christopher Akritidis Apr 27 '13 at 7:45

If you take, for example, a perfect metal sphere then it has a work function that is the energy required to remove an electron from the metal to infinity. If you start charging the sphere by adding electrons to it then the work function decreases, and above some limiting charge the work function falls to zero. This means any more electrons you add to the sphere immediately escape again. This is an example of a phenomenon is called field emission.

I've chosen the example of a metal sphere since it's nice and simple, but this will apply to any object, and it means that there is a maximum charge that can be sustained on any object regardless of how close to perfectly it has been made.

If you keep the charge below the level where field emission occurs, and keep the object in a vacuum, and mask it from any light with energy of greater than the work function, and keep it at a temperature below which thermionic emission occurs, then the sphere will stay charged forever.

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Thank you John, your answer definitely sets me on the correct path. – Christopher Akritidis Apr 27 '13 at 14:52
Based on John’s answer, I found the following calculator: pulsedpower.net/Applets/pulsedpower/fieldemission/… I can’t vouch for its accuracy, but I entered the following: Electric Field of Conducting Sphere E = Q/(4*π*r^2*ε0) = D/ε0 ε0 = ~10^-11 F/m is the vacuum permittivity => E = ~10^5/10^-11 V/m = ~10^16V/m (= 10^8MV/cm) Surface is 4πR^2 = 10*100μm^2 =1000μm^2 The electron wave function of metals is in the order of eV The result was that the field emission effect was way too high for ordinary electron wave functions. – Christopher Akritidis Apr 27 '13 at 16:23
"mask it from any light" will of course have to include "any form of radiation". The discharging of objects has been used to determine the presence of ionizing radiation (although usually in the presence of a gas). And your vacuum has to be pretty good too... Is there a temperature below which there is no thermionic emission, or does the number just become "very small"? – Floris Sep 4 '14 at 12:52
field emission seems to limit maximum negative charge densities, but it seems to me that the equivalent of the work function for ions in a positively charged metal surface would be vastly superior than that given by the electron work function – diffeomorphism Dec 15 '14 at 19:39

One would expect that a positively charged sphere could hold a different amount of charge than a negatively charged one, one reason being that the tunneling probability for electrons to escape would tend to be different than for ions. There may be other effects as well that could cause a difference. If I were to venture a guess I would suppose that more positive charge density could be stored on a sphere than negative charge density. Field emission of electrons is a standard type of electron gun, but for ion sources one usually hits the surface with an energetic particle to cause the ions to jump out, as in sputtering. Without incident energetic particles, the ions tend to stay put, so I suspect that more positive charge could be stored than negative.

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